Numerical methods in molecular dynamics T. Lelivre CERMICS - Ecole - PowerPoint PPT Presentation

Introduction Adaptive biasing techniques The Parallel Replica Algorithm Conclusion Numerical methods in molecular dynamics T. Lelièvre CERMICS - Ecole des Ponts ParisTech & MicMac project-team - INRIA Joint work with C. Chipot, C. Le Bris, M. Luskin, K. Minoukadeh, D. Perez, M. Rousset and G. Stoltz. GDR CHANT, Grenoble November 2011

Introduction Adaptive biasing techniques The Parallel Replica Algorithm Conclusion Introduction The aim of molecular dynamics simulations is to understand the relationships between the macroscopic properties of a molecular system and its atomistic features. In particular, one would like to to evaluate numerically macroscopic quantities from models at the microscopic scale. Some examples of macroscopic quantities: (i) Thermodynamics quantities (average of some observable wrt an equilibrium measure): stress, heat capacity, free energy,... � E µ ( ϕ ( X )) = R d ϕ ( x ) µ ( d x ) . (ii) Dynamical quantities (average over trajectories at equilibrium): diffusion coefficients, viscosity, transition rates,... � E ( F (( X t ) t ≥ 0 )) = F (( x t ) t ≥ 0 )) W ( d (( x t ) t ≥ 0 )) . C 0 ( R + , R d )

Introduction Adaptive biasing techniques The Parallel Replica Algorithm Conclusion Introduction Many applications in various fields: biology, physics, chemistry, materials science. Molecular dynamics computations consume today a lot of CPU time. A molecular dynamics model amounts essentially in choosing a potential V which associates to a configuration ( x 1 , ..., x N ) = x ∈ R 3 N an energy V ( x 1 , ..., x N ) . In the canonical (NVT) ensemble, configurations are distributed according to the Boltzmann-Gibbs probability measure: d µ ( x ) = Z − 1 exp ( − β V ( x )) d x , � where Z = exp ( − β V ( x )) d x is the partition function and β = ( k B T ) − 1 is proportional to the inverse of the temperature.

Introduction Adaptive biasing techniques The Parallel Replica Algorithm Conclusion Introduction Typically, V is a sum of potentials modelling interaction between two particles, three particles and four particles: � � � V = V 1 ( x i , x j ) + V 2 ( x i , x j , x k ) + V 3 ( x i , x j , x k , x l ) . i < j i < j < k i < j < k < l For example, V 1 ( x i , x j ) = V LJ ( | x i − x j | ) where �� σ � 12 − � σ � 6 � V LJ ( r ) = 4 ǫ is the Lennard-Jones potential. r r Difficulties: (i) high-dimensional problem ( N ≫ 1) ; (ii) µ is a multimodal measure.

Introduction Adaptive biasing techniques The Parallel Replica Algorithm Conclusion Introduction To sample µ , ergodic dynamics wrt to µ are used. A typical example is the over-damped Langevin (or gradient) dynamics: � 2 β − 1 d W t . d X t = −∇ V ( X t ) dt + It is the limit (when the mass goes to zero or the damping parameter to infinity) of the Langevin dynamics : ( d X t = M − 1 P t dt , d P t = −∇ V ( X t ) dt − γ M − 1 P t dt + p 2 γβ − 1 d W t , where M is the mass tensor and γ is the friction coefficient. To compute dynamical quantities, these are also typically the dynamics of interest. Thus, � T N E µ ( ϕ ( X )) ≃ 1 ϕ ( X t ) dt and E ( F (( X t ) t ≥ 0 )) ≃ 1 � F (( X m t ) t ≥ 0 ) . T N 0 m = 1 In the following, we mainly consider the over-damped Langevin dynamics.

Introduction Adaptive biasing techniques The Parallel Replica Algorithm Conclusion Introduction Difficulty: In practice, X t is a metastable process, so that the convergence to equilibrium is very slow. A 2d schematic picture: X 1 t is a slow variable (a metastable dof) of the system. x 2 V ( x 1 , x 2 ) X 1 t x 1 t

Introduction Adaptive biasing techniques The Parallel Replica Algorithm Conclusion Introduction A more realistic example (Dellago, Geissler) : Influence of the solvation on a dimer conformation. �� Compact state. Stretched state. The particles interact through a pair potential: truncated LJ for all particles except the two monomers (black particles) which interact through a double-well potential. A slow variable is the distance between the two monomers.

Introduction Adaptive biasing techniques The Parallel Replica Algorithm Conclusion Introduction One central numerical difficulty is thus metastability. Outline of the talk: 1. Adaptive biasing techniques: These belong to one class of numerical methods to compute thermodynamic quantities, and in particular free energy differences. 2. The Parallel Replica dynamics: This is one instance of an algorithm to generate efficiently metastable dynamics.

Introduction Adaptive biasing techniques The Parallel Replica Algorithm Conclusion Adaptive biasing techniques

Introduction Adaptive biasing techniques The Parallel Replica Algorithm Conclusion Adaptive biasing techniques We suppose in this part that we know a slow variable of dimension 1: ξ ( X t ) , where ξ : R d → T is a so-called reaction coordinate. This reaction coordinate will be used to bias the dynamics (adaptive importance sampling technique). For example, in the 2d simple system, ξ ( x 1 , x 2 ) = x 1 . x 2 V ( x 1 , x 2 ) X 1 t x 1 t

Numerical methods in molecular dynamics T. Lelivre CERMICS - Ecole - PowerPoint PPT Presentation

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Numerical methods to overcome metastability in molecular dynamics T. Lelivre CERMICS - Ecole

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